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| outer automorphism groups of free groups | |
|---|---|
| Name | Outer automorphism groups of free groups |
| Othernames | Out(F_n) |
| Field | Algebraic topology, Geometric group theory |
| Notable | Bestvina–Feighn, Culler–Vogtmann, Thurston |
outer automorphism groups of free groups
Outer automorphism groups of free groups are the groups of equivalence classes of automorphisms of free groups under inner automorphisms. They arose in the work of Andreas G. Nielsen, Jakob Nielsen, and were developed by Hyman Bass, John H. Conway, Marc Culler, Karen Vogtmann, Mladen Bestvina, and Mark Feighn; they play central roles in the interplay between William Thurston's theory of surface mapping classes, Serre's theory of trees, and problems in Gromov's hyperbolic groups and Sela's rigidity theory.
The study of Out(F_n) for finite rank n connects classical work of Nielsen and Jakob Nielsen on surface mappings with modern results by Bestvina, Feighn, Handel, and Mosher. For n = 2, Out(F_2) is isomorphic to GL(2,Z), linking to Modular group phenomena studied by Farey, Gauss, and Dedekind. For n ≥ 3, Out(F_n) exhibits rich behavior analogous to Mapping class group properties of surfaces studied by Thurston and Harvey, while also relating to rigidity results of Mostow and Margulis.
Let F_n denote the free group of rank n on generators a_1,...,a_n; its automorphism group Aut(F_n) was studied by Nielsen and W. Magnus. The outer automorphism group Out(F_n) = Aut(F_n)/Inn(F_n) quotients by inner automorphisms generated by conjugation elements studied in Dehn's and Baer's early work. Out(F_n) contains analogues of Dehn twist-type elements, has subgroups isomorphic to GL(n,Z), and displays subgroup phenomena explored by Gaboriau and Levitt.
Outer space CV_n, introduced by Culler and Vogtmann, is a contractible space on which Out(F_n) acts properly discontinuously with finite stabilizers, paralleling Teichmüller space for Mapping class group. Points of CV_n correspond to marked metric graphs related to work by Stallings on graphs of groups and to Serre's theory of trees. Compactifications of CV_n use projective length functions studied by Paulin and connections to Morgan and Shalen's R-tree theory; these compactifications were expanded by Bestvina and Feighn and inform analogues of Thurston compactification.
Out(F_n) has a rich lattice of subgroups including stabilizers of free factor systems and free splittings investigated by Guirardel, Levitt, and Handel. The Tits alternative for Out(F_n), proved by Bestvina and Feighn and refined by Bridson and Vogtmann, parallels results for Linear groups like Tits's theorem and for Mapping class groups by Ivanov. Centralizers, normalizers, and virtually abelian subgroups were analyzed by Bogopolski, McCool, and Cohen; residual finiteness and decision problems connect to work of Kapovich and Myasnikov.
Train track maps for elements of Out(F_n) were introduced by Bestvina and Handel adapting ideas of Thurston and Bestvina–Feighn. Fully irreducible ("iwip") elements generalize pseudo-Anosov maps of Fathi, Laudenbach, and Poénaru and were studied in depth by Levitt, Lustig, and Feighn. North–South dynamics on compactified Outer space mirror dynamics for Pseudo-Anosov classes in Teichmüller theory; attracting and repelling trees connect to work by Skora and Guirardel–Levitt.
Cohomological studies of Out(F_n) use equivariant homology of CV_n and spectral sequences developed by Brown and Serre. Virtual cohomological dimension computations were established by Culler and Vogtmann and refined by Hatcher and Vogtmann; stability phenomena relate to homological stability results by Hatcher–Vogtmann and have analogues in Harer's stability for mapping class groups. Finiteness properties (type F_n, type FP) for subgroups and for Out(F_n) itself involve work of Levitt, Brady, and Bestvina–Feighn and connect to algorithmic problems studied by Gersten and Stallings.
Out(F_n) interfaces with many areas: mapping class groups of surfaces (work of Thurston, Harvey), automorphism groups of right-angled Artin groups analyzed by Charney and Vogtmann, and arithmetic groups like GL(n,Z). Applications to 3-manifold topology draw on Thurston and Perelman, while connections to Gromov's hyperbolic groups and Sela's JSJ decomposition theory inform rigidity and classification results. Dynamics on character varieties relate to Goldman and Morgan–Shalen; algorithmic and complexity aspects tie to Makanin and Razborov.